$1/d$ Expansion for $k$-Core Percolation

Abstract

The physics of $k$-core percolation pertains to those systems whose constituents require a minimum number of $k$ connections to each other in order to participate in any clustering phenomenon. Examples of such a phenomenon range from orientational ordering in solid ortho-para ${\mathrm{H}}_2$ mixtures to the onset of rigidity in bar-joint networks to dynamical arrest in glass-forming liquids. Unlike ordinary ($k=1$) and biconnected ($k=2$) percolation, the mean field $k\ge 3$-core percolation transition is both continuous and discontinuous, i.e. there is a jump in the order parameter accompanied with a diverging length scale. To determine whether or not this hybrid transition survives in finite dimensions, we present a $1/d$ expansion for $k$-core percolation on the $d$-dimensional hypercubic lattice. We show that to order $1/d^3$ the singularity in the order parameter and in the susceptibility occur at the same value of the occupation probability. This result suggests that the unusual hybrid nature of the mean field $k$-core transition survives in high dimensions.

47 pages, 26 figures, revtex4

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Cited by in corpus (10)

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• Critical behavior of $k$-core percolation: Numerical studies
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• The fate of the bootstrap percolation hybrid critical point in finite dimension
• On $k$-Core Percolation in Four Dimensions
• Dynamics of k-core percolation in a random graph
• Dynamics of k-core percolation
• Quantum $k$-core conduction on the Bethe lattice
• Constraint percolation on hyperbolic lattices
• Level statistics for quantum $k$-core percolation